CIE May 2020 9709 Pure Maths Paper 23

CIE May/June 2020 9709 Pure Maths Paper 23 (pdf)

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1. Given that 2^y = 9^3x, use logarithms to show that y = kx and find the value of k correct to 3 significant figures.
   
2. Find the exact coordinates of the stationary point on the curve with equation y = 5xe^1/2x.
3. The equation of a curve is cos 3x + 5 sin y = 3.
   Find the gradient of the curve at the point (1/9π,1/6π).
4. The variables x and y satisfy the equation y = Ax^−2p, where A and p are constants. The graph of ln y against ln x is a straight line passing through the points (−0.68, 3.02) and (1.07, −1.53), as shown in the diagram.
   Find the values of A and p.
5. (a) Sketch, on the same diagram, the graphs of y = |2x − 3| and y = 3x + 5.
   (b) Solve the inequality 3x + 5 < |2x − 3|.

6. The polynomial p(x) is defined by
   p(x) = 6x³ + ax² − 4x − 3,
   where a is a constant. It is given that (x + 3) is a factor of p(x).
   (a) Find the value of a.
   (b) Using this value of a, factorise p(x) completely.
   (c) Hence solve the equation p(cosec θ) = 0 for 0° < 1 < 360°.
7. It is given that
   (a) Show that a =
   (b) Using the equation in part (a), show by calculation that 1 < a < 2.
   (c) Use an iterative formula, based on the equation in part (a), to find the value of a correct to 4 significant figures. Give the result of each iteration to 6 significant figures.
8. (a) Show that 3 sin 2θ cot θ = 6 cos²θ.
   (b) Solve the equation 3 sin 2θ cot θ = 5 for 0 < 1 < π.
   (c) Find the exact value of

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